Once we have at our disposal the infrared-safe PWAs we proceed to its unitarization by employing standard techniques in hadron physics. In this way, for calculating the unitarized PWA \(T^{(J)}_{\lambda _3\lambda _4,\lambda _1\lambda _2}(s)\)  the unitary or right-hand cut can be resummed by employing the general expression


2||/4T(J)34,12(s)=[12||/4V34,12(J)+18-s2]-1 ,

where \(\Lambda \)  is the cutoff of the low-energy gravity EFT. This expression arises by performing a dispersion relation of the inverse of the PWA with a circle in infinity deformed to engulf the right-hand cut [1]}. In the subsequent, we identify \(\Lambda \)  with the unitary cutoff \(\Lambda _U^2=\pi /(G \ln a)\)  and look for poles of \(T^{(0)}_{22,22}(s)\)  in the second RS, which implies the following secular equation for the pole positions,


1x+(-x)-i2=0 , x=sP2 .

It can be easily solved by iteration or numerically, \(x=0.07-i\,0.20\simeq -i2/(3\pi )\) . The estimated uncertainty on \(x\)  from higher order contributions is a 20% [2]}, [1]}. The reason is because the main source of expected uncertainty stems from the one-loop or NLO graviton-graviton PWA, which has a size that scales as \(|x|=|s_P|/\Lambda _U^2\) .
